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Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane

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Abstract

We study α-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for α ∈ (1,2). We describe the corresponding Hardy spaces and prove the Fatou theorem for α-harmonic functions. We also give explicit formulas for the Martin kernel of the complement of the sphere and for the harmonic measure, Green function and Martin kernel of the complement of the hyperplane for the symmetric α-stable Lévy processes. Some extensions for the relativistic α-stable processes are discussed.

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References

  1. Aikawa, H.: Boundary Harnack principle and Martin boundary for a uniform domain. J. Math. Soc. Jpn. 53(1), 119–145 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier (Grenoble) 28(4), 169–213 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, New York (2001)

    MATH  Google Scholar 

  4. Bass, R.F., You, D.: A Fatou theorem for α-harmonic functions. Bull. Sci. Math. 127(7), 635–648 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Stud. Math. 123, 43–80 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Bogdan, K.: Representation of α-harmonic functions in Lipschitz domains. Hiroshima Math. J. 29, 227–243 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Stud. Math. 133, 53–92 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondraček, Z.: Potential Analysis of Stable Processes and Its Extensions. Lecture Notes in Mathematics, vol. 1980. Springer, Berlin (2009)

    Book  Google Scholar 

  9. Bogdan, K., Dyda, B.: Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains. Stud. Math. (MR 1 980 119) 157, 83–96 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bogdan, K., Dyda, B., Luks, T.: On Hardy spaces. Available at arxiv: 1109.0210 (2011). Accessed 1 Sep 2011

  11. Bogdan, K., Jakubowski, T.: Probleme de Dirichlet pour les fonctions α-harmoniques sur les domaines coniques. Ann. Math. Blaise Pascal 12, 297–308 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of α-harmonic functions. Probab. Theory Relat. Fields 140(3–4), 345–381 (2008)

    MATH  Google Scholar 

  13. Bogdan, K., Żak, T.: On Kelvin transformation. J. Theor. Probab. 19(1), 89–120 (2006)

    Article  MATH  Google Scholar 

  14. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion - Facts and Formulae, 2nd edn. Birkhäuser Verlag, Basel (2002)

    Book  MATH  Google Scholar 

  15. Byczkowski, T., Małecki, J., Ryznar, M.: Bessel potentials, hitting distributions and Green functions. Trans. Am. Math. Soc. 361(9), 4871–4900 (2009)

    Article  MATH  Google Scholar 

  16. Byczkowski, T., Małecki, J., Ryznar, M.: Hitting Half-spaces by Bessel-Brownian diffusions. Potential Anal. 33(1), 47–83 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Caffarelli, L.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12(5), 1151–1179 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, Z.-Q., Kim, P., Song, R.: Green function estimates for relativistic stable processes in half-space-like open sets. Stoch. Process. Their Appl. 121(5), 1148–1172 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, Z.-Q., Song, R.: Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159(1), 267–294 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, Z.-Q., Song, R.: Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201(1), 262–281 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Doob, J.L.: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York (1984)

    Book  MATH  Google Scholar 

  22. Duren, P.L.: Theory of H p Spaces. Academic Press, New York (1970); Reprinted with Supplement by Dover Publications, Mineola, NY (2000)

    MATH  Google Scholar 

  23. Erdelyi, A.: Higher Transcendental Functions. Bateman Manuscript Project, vol. I, II. McGraw-Hill, New York (1953)

    Google Scholar 

  24. Gerhold, S.: Asymptotics for a variant of the Mittag-Leffler function. Integral Transforms Spec. Funct. 23(6), 397–403 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grzywny, T., Ryznar, M.: Estimates of Green function for some perturbations of fractional Laplacian. Ill. J. Math. 51(4), 1409–1438 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Grzywny, T., Ryznar, M.: Two-sided optimal bounds for half-spaces for relativistic α-stable process. Potential Anal. 28(3), 201–239 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Isozaki, Y.: Hitting of a line or a half-line in the plane by two-dimensional symmetric stable Lévy processes. Stoch. Process. Their Appl. 121, 1749–1769 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jerison, D.S., Kenig, C.E.: Boundary value problems on Lipschitz domains. In: Studies in Partial Differential Equations, MAA Stud. Math., Math. Assoc, vol. 23, pp. 1–68, America, Washington, DC (1982)

  29. Kim, P.: Relative Fatou’s theorem for ( − Δ)α/2-harmonic functions in bounded κ-fat open sets. J. Funct. Anal. 234(1), 70–105 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kim, P., Lee, Y.-R.: Generalized 3G theorem and application to relativistic stable process on non-smooth open sets. J. Funct. Anal. 246(1), 113–143 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle for subordinate Brownian motions. Stoch. Process. Their Appl. 119(5), 1601–1631 (2009)

    Article  MATH  Google Scholar 

  32. Koosis, P.: Introduction to H p Spaces. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  33. Kulczycki, T., Siudeja, B.: Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes. Trans. Am. Math. Soc. 358(11), 5025–5057 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Luks, T.: Hardy spaces for the Laplacian with lower order perturbations. Stud. Math. 204, 39–62 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Michalik, K., Ryznar, M.: Relative Fatou theorem for α-harmonic functions in Lipschitz domains. Ill. J. Math. 48(3), 977–998 (2004)

    MathSciNet  MATH  Google Scholar 

  36. Michalik, K., Ryznar, M.: Hardy spaces for α-harmonic functions in regular domains. Math. Z. 265(1), 173–186 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Michalik, K., Samotij, K.: Martin representation for α-harmonic functions. Probab. Math. Stat. 20, 75–91 (2000)

    MathSciNet  MATH  Google Scholar 

  38. Port, S.C.: Hitting times and potential for recurrent stable processes. J. Anal. Math. 20, 371–395 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  39. Port, S.C.: The first hitting distribution of a sphere for symmetric stable processes. Trans. Am. Math. Soc. 135, 115–125 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ryznar, M.: Estimates of Green function for relativistic α-stable process. Potential Anal. 17(1), 1–23 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Song, R., Wu, J.-M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168, 403–427 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  43. Stein, E.M.: Boundary Behavior of Holomorphic Functions of Several Complex Variables. Princeton University Press and University of Tokyo Press, Princeton, NJ (1972)

    MATH  Google Scholar 

  44. Widman, K.O.: On the boundary behavior of solutions to a class of elliptic partial differential equations. Ark. Mat. 6, 485–533 (1966)

    Article  MathSciNet  Google Scholar 

  45. Wu, J.-M.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier Grenoble 28, 147–167 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wu, J.-M.: Harmonic measures for symmetric stable processes. Stud. Math. 149, 281–293 (2002)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. LAREMA, Laboratoire de Mathématiques, Université d’Angers, 2 boulevard Lavoisier, 49045, Angers Cedex 01, France

    Tomasz Luks

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  1. Tomasz Luks
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Correspondence to Tomasz Luks.

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This research was partially supported by Agence Nationale de la Recherche grant ANR-09-BLAN-0084-01 and by MNiSW grant N N201 373136.

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Luks, T. Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane. Potential Anal 39, 29–67 (2013). https://doi.org/10.1007/s11118-012-9321-x

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